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Properties

Label 232.4.k.a
Level $232$
Weight $4$
Character orbit 232.k
Analytic conductor $13.688$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,4,Mod(75,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.75");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.k (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.6884431213\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(88\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 4 q^{2} - 4 q^{3} + 26 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 4 q^{2} - 4 q^{3} + 26 q^{8} - 18 q^{10} + 36 q^{11} + 130 q^{12} - 56 q^{14} + 296 q^{16} - 56 q^{17} - 266 q^{18} - 4 q^{19} - 260 q^{20} + 176 q^{24} + 3904 q^{25} - 98 q^{26} - 160 q^{27} - 1708 q^{30} + 246 q^{32} + 1272 q^{36} - 830 q^{40} - 240 q^{41} - 812 q^{43} + 2014 q^{44} + 852 q^{46} + 1090 q^{48} - 7456 q^{49} + 278 q^{50} - 296 q^{52} + 2480 q^{54} - 1392 q^{56} + 1028 q^{58} - 8 q^{59} - 1414 q^{60} - 1008 q^{65} + 398 q^{66} - 408 q^{68} - 2520 q^{70} + 568 q^{72} + 736 q^{73} + 5700 q^{74} + 2692 q^{75} - 884 q^{76} + 752 q^{78} - 14136 q^{81} + 1456 q^{82} - 8 q^{83} - 2064 q^{84} + 2548 q^{88} + 1064 q^{89} + 1220 q^{90} + 6332 q^{94} + 576 q^{97} + 2136 q^{98} + 1220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −2.82843 − 0.00120906i −2.16238 + 2.16238i 8.00000 + 0.00683948i −0.745053 6.11873 − 6.11351i − 31.3758i −22.6274 − 0.0290174i 17.6483i 2.10733 0.000900813i
75.2 −2.82765 − 0.0661740i 2.56631 − 2.56631i 7.99124 + 0.374234i −12.7141 −7.42645 + 7.08680i 8.41253i −22.5717 − 1.58702i 13.8281i 35.9512 + 0.841346i
75.3 −2.82752 − 0.0715985i 4.82050 − 4.82050i 7.98975 + 0.404893i 18.5153 −13.9752 + 13.2849i − 21.0291i −22.5622 − 1.71690i − 19.4745i −52.3524 − 1.32567i
75.4 −2.82130 − 0.200607i −6.00898 + 6.00898i 7.91951 + 1.13195i 10.6669 18.1586 − 15.7477i − 4.06679i −22.1163 − 4.78228i − 45.2156i −30.0945 − 2.13985i
75.5 −2.78217 + 0.509429i 3.41908 − 3.41908i 7.48096 − 2.83464i 6.44780 −7.77068 + 11.2542i 10.9654i −19.3693 + 11.6975i 3.61982i −17.9389 + 3.28469i
75.6 −2.75890 + 0.623267i −5.28232 + 5.28232i 7.22308 − 3.43906i −11.0406 11.2811 − 17.8657i − 0.427038i −17.7843 + 13.9899i − 28.8058i 30.4601 − 6.88127i
75.7 −2.75026 − 0.660335i 1.18481 − 1.18481i 7.12791 + 3.63219i −1.52127 −4.04090 + 2.47616i − 5.67460i −17.2052 − 14.6963i 24.1925i 4.18390 + 1.00455i
75.8 −2.71455 + 0.794490i −2.13769 + 2.13769i 6.73757 − 4.31337i 19.1413 4.10450 − 7.50125i 17.6779i −14.8625 + 17.0618i 17.8605i −51.9600 + 15.2076i
75.9 −2.70424 + 0.828900i 7.11990 − 7.11990i 6.62585 − 4.48309i 2.13042 −13.3522 + 25.1556i 7.02494i −14.2019 + 17.6155i − 74.3859i −5.76116 + 1.76590i
75.10 −2.65607 + 0.972250i −2.58243 + 2.58243i 6.10946 − 5.16474i −16.7003 4.34836 − 9.36990i 32.2272i −11.2058 + 19.6578i 13.6621i 44.3573 − 16.2369i
75.11 −2.65312 − 0.980281i −1.28740 + 1.28740i 6.07810 + 5.20161i 6.00063 4.67764 − 2.15361i 34.3388i −11.0269 − 19.7587i 23.6852i −15.9204 − 5.88231i
75.12 −2.55878 + 1.20525i 4.22460 − 4.22460i 5.09476 − 6.16794i −21.2219 −5.71815 + 15.9015i − 34.5918i −5.60251 + 21.9229i − 8.69442i 54.3023 − 25.5776i
75.13 −2.53752 − 1.24940i −3.49118 + 3.49118i 4.87798 + 6.34077i −18.7575 13.2208 − 4.49703i − 1.97452i −4.45577 − 22.1844i 2.62331i 47.5975 + 23.4357i
75.14 −2.47198 − 1.37452i 6.27275 − 6.27275i 4.22136 + 6.79559i −14.8256 −24.1282 + 6.88406i 15.3608i −1.09442 − 22.6009i − 51.6948i 36.6486 + 20.3782i
75.15 −2.44754 − 1.41758i 4.68373 − 4.68373i 3.98091 + 6.93919i 14.3342 −18.1032 + 4.82405i 14.8348i 0.0934373 − 22.6272i − 16.8747i −35.0836 − 20.3199i
75.16 −2.42201 + 1.46078i −1.34911 + 1.34911i 3.73226 − 7.07603i −0.225176 1.29681 − 5.23831i − 10.7256i 1.29692 + 22.5902i 23.3598i 0.545379 − 0.328932i
75.17 −2.41579 + 1.47104i 0.0916324 − 0.0916324i 3.67209 − 7.10744i 14.3095 −0.0865700 + 0.356160i − 9.92961i 1.58431 + 22.5719i 26.9832i −34.5689 + 21.0499i
75.18 −2.41577 − 1.47107i −6.84832 + 6.84832i 3.67189 + 7.10755i −6.62987 26.6184 − 6.46959i 16.5367i 1.58528 − 22.5718i − 66.7991i 16.0162 + 9.75302i
75.19 −2.34524 − 1.58109i 5.11166 − 5.11166i 3.00029 + 7.41608i 2.29147 −20.0701 + 3.90606i − 33.9381i 4.68910 − 22.1362i − 25.2581i −5.37403 − 3.62302i
75.20 −2.24235 − 1.72391i −3.48342 + 3.48342i 2.05625 + 7.73122i 15.3375 13.8161 − 1.80593i − 17.1191i 8.71712 − 20.8809i 2.73161i −34.3920 − 26.4405i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
29.c odd 4 1 inner
232.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.4.k.a ✓ 176
8.d odd 2 1 inner 232.4.k.a ✓ 176
29.c odd 4 1 inner 232.4.k.a ✓ 176
232.k even 4 1 inner 232.4.k.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.k.a ✓ 176 1.a even 1 1 trivial
232.4.k.a ✓ 176 8.d odd 2 1 inner
232.4.k.a ✓ 176 29.c odd 4 1 inner
232.4.k.a ✓ 176 232.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).